An uncertainty inequality for groups of order \(pq\) (Q1199777)

Let \(G\) be a finite group. For a function \(f: G\to \mathbb{C}\) and a representation \(\rho : G \to GL(V)\) of \(G\) the endomorphism \(\widehat{f}(\rho) = \sum_{x \in G}f(x)\rho(x)\) of \(V\) is called the Fourier transform of \(f\) at \(\rho\). If \(\rho_ 1,\rho_ 2,\dots,\rho_ t\) denote the complex irreducible representations of \(G\), where \(\rho_ i: G\to GL(V_ i)\), and \(\text{deg }\rho_ i = \dim V_ i = n_ i\), then define \(\mu(f) = \sum^ t_{i = 1}\dim V_ i\cdot \text{rank }\widehat{f}(\rho_ i)\). The main result of this paper is theorem 1: Let \(0 \neq f : G \to \mathbb{C}\). Then (a) \(| \text{Supp }f| \mu(f) \geq | G|\) (b) Suppose \(f(1) = 1\). Then \(| \text{Supp}(f)| \mu(f) = | G|\) if and only if \(H = \text{Supp }f\) is a subgroup of \(G\), and \(f(x) = 1_ H(x)\chi(x)\) where \(\chi\) is a 1-dimensional character of \(G\), and \(1_ H(x)\) is the indicator function of \(H\subseteq G\). As an application of this theorem the author obtains an uncertainty-type inequality for direct products of non-abelian groups of order \(pq\) where \(p\) and \(q\) are prime numbers and \(p\mid q - 1\).